Problems with Randomization .

Problems with Randomization .
Andre Willers
16 Nov 2010

Synopsis :
Any random number of delineated items can be described as a non-random set .
What appears random , is not .

Discussion :
See http://andreswhy.blogspot.com "Randomness2" et al .

A quick derivation:
Let a delineated set of elements A() have Aleph(0) elements , ie countable .
Aleph(0) : A(1),A(2), A(3), … , A(Aleph(0) ) with n=Aleph(0) elements .

The Powerset of this is all the possible combinations C of the set of n elements .
This can be counted as : nC0 + nC1 + nC2 + … nCn = 2^n (Binomial expansion)
Where nCk=n! / { (n-k)! * k! }
But the PowerSet has (n+1) terms .

Now , pair each term nCk with a term in Aleph(0) set . We ignore SuperCounting . There will always be one left over . This means that the Powerset cannot be counted in terms of Aleph(0) . It must then be of a higher order of infinity . ie Aleph(1) .

This is iterated to give higher orders of Aleph .

Important note :
The meaning of repetition of terms in the basal Aleph(0) set .
Each successive Combinations involves the original terms . This denotes feedback processes or causal relationships .
Think feedback descriptions without the time-dimension .
They clump together .


Randomness :
Let Beth(m) denote the Order of Randomness of Aleph(m) .

Any finite selection of once-only combination of terms in Aleph(0) can be expressed as a term in Aleph(1) . (By definition)

If there can be more than repetition of an Aleph(0) term , then we can always (by definition) find a Aleph(x) term that incorporates these terms . But the true randomness is then expressed only relative to this particular term .
This can be exactly calculated , since the Aleph(0) items are delineated .
The numbers can be large , but ratio's and countability can be used .

The problem :
Randomization in medicine , marketing , etc is routinely used with naïve randomization on Aleph(0) items . But these are never random . Results can be extremely misleading .

How to choose a truly random set :
We use the trusty old diagonal proof :
Write the transfinite set as
Aleph(0) : A(1),A(2), A(3), … , A(Aleph(0) )
Aleph(1) : S(1,1) , S(1,2) , …
Aleph(2) : S(2,1) , S(2,2) , …
Continue…
Aleph(w) : S(w,1) , S(w,2) , …

Where S(j,z) is Powerset of previous S(j-1),k)
Now do a change on each element of the diagonal and designate this diagonal set as Aleph(w+1) . This cannot fit any previous Aleph (by definition of diagonal Process)
This is an iterative , transfinite process , but it can be truncated at any stage to give a randomness with exactly calculated , finite boundaries .
(This is done all the time in Quantum Physics , to avoid those pesky infinities.)
What does all this mean ?
Sigh . All those nice little studies using random assignations of groups without taking into account feedback processes will have to be redone . At best , they are meaningless . At worst , they are harmful , giving false positives or negatives .
It is like doing a gene-analysis , and ignoring all the gene-repetitions . You see ?
Beth has many rooms .
Andre